Universality of multiplicative infinite loop space machines

Abstract

We establish a canonical and unique tensor product for commutative monoids and groups in an $\infty$–category $\mathcal{C}$ which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that ${\mathbb{E}}_n$–(semi)ring objects in $\mathcal{C}$ give rise to ${\mathbb{E}}_n$–ring spectrum objects in $\mathcal{C}$. In the case that $\mathcal{C}$ is the $\infty$–category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic $K$–theory of rings and ring spectra.

The main tool we use to establish these results is the theory of smashing localizations of presentable $\infty$–categories. In particular, we identify preadditive and additive $\infty$–categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show $Ring\left(\mathcal{D}\otimes \mathcal{C}\right)\simeq Ring\left(\mathcal{D}\right)\otimes \mathcal{C}$. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in $\infty$–categories.